Lecture09A - Lecture 9 1 Partition Functions for Molecules...

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Unformatted text preview: Lecture 9 1 Partition Functions for Molecules (18.3-18.8) Moving beyond atoms to molecules, we have two more independent types of degrees of freedom: vibration and rotation. trans rot vib elec = + + + ( 29 and accordingly: q V,T trans rot vib elec q q q q = ( 29 ( 29 , and, as always: Q N,V,T ! N q V T N = First, consider a diatomic molecule: ( 29 ( 29 3/ 2 1 2 2 2 , B trans m m k T q V T V h + = Where the total mass of the molecule (m 1 +m 2 ) is substituted for m in our monatomic solution. Lecture 9 2 For the electronic energy contribution for a diatomic, we need to decide what will be our zero-energy reference point. Relative to their constituent atoms, diatomics have electronic energy. Fig. 18.2 We will define separated atoms at R= in their ground electronic state as our zero for energy. 2 / / 1 2 : ... e B e B D k T k T elec e e Thus q g e g e - = + + Lecture 9 3 Now we can consider vibrations . v 1 v v 0,1,2... for a H.O. 2 hv = + = Note: in your text, the vs (quantum numbers) look a lot like the v s (frequencies). Be careful! ( 29 ( 29 v v 1/2 /2 v v 0 v 0 v 0 hv hv hv vib q T e e e e - +--- = = = = = = 2 1 for 1: 1 ... 1 n n x x x x x = < = + + + =- See the back cover of your text for common expansions. ( 29 / 2 since 1, 1 hv hv vib hv e e q T e --- < =- Wow! Simple. Statistical mechanicians (mechanics?) like to simplify by using single terms (like =1/k B T), so they define: vib B hv k = vib This relates the vibrational frequency, , to a vibrational temperature, . v ( 29 / 2 / 1 vib vib T vib T e q T e-- =- Lecture 9 4 2 vib ln As we noted before: E vib B d q Nk T dT = / 2 1 vib vib vib B T Nk e = + - As I showed in lecture 6: ( 29 2 / , , 2 / 1 vib vib T V vib vib V vib T C e C R n T e-- = = - , , V vib V vib as T C as T C R ( 29 ( 29 1 , 2 2 1 0.368 , 0.921 0.632 1 vib V vib e at T C R R R e-- = = = =- That is, at the vibrational temperature, the heat capacity is 92.1% of the high temperature limit. Lecture 9 5 We can also use these partition functions to calculate the population distribution as a function of temperature. Since all ideal diatomic gases will behave the same, we will consider temperatures in terms of vib ....
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This note was uploaded on 04/08/2010 for the course CHEM 444 taught by Professor Jameslis during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Lecture09A - Lecture 9 1 Partition Functions for Molecules...

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